Can someone define what a "run book" is? Previous experience gives me the impression that the general term for "run book" is documentation pertaining to steps required for a task, such as steps to run a database job, how to reboot a system, update a system with patches. This book should be detailed enough so that anyone can do a procedure regardless is she knew the system well.

As with all documentation, you could define each as a CI in configuration management I suppose (although if you are maintaining a knowledge base that includes these then I don't think I'd record each as a CI as it's duplication of work).

In the context of Service Management, a Run Book is a set of directions for what to do when you encounter an Incident, as related to a Product or Service. For example, if someone calls the Service Desk with a statement that they're seeing an error message "XYZ" while using product "ABC". The Service Desk would open up the Run Book for product ABC and look for directions on what to do if such an error message is encountered.

Run Books tend to be most common when dealing with scheduled batch processing systems. So, for example, if a batch stream fails at a specific point, the Run Book will tell you what needs to be done to address the failure and get the stream back in progress. However, even so, it is still a support document.

In the context of Service Management, a Run Book is a set of directions for what to do when you encounter an Incident, as related to a Product or Service. For example, if someone calls the Service Desk with a statement that they're seeing an error message "XYZ" while using product "ABC". The Service Desk would open up the Run Book for product ABC and look for directions on what to do if such an error message is encountered

I agree to the above and the Run Book falls under Incident Management under the ITIL components

I'm going to break away here. I feel that a Run Book is actual part of Problem Management as it contains the Known Errors on the batch job. However, Incident Management interacts with Problem Management and would find the information through referencing the Known Error Database and finding the Run Book.